L(s) = 1 | + 2.85·5-s + 1.85·7-s − 3.23·13-s − 4·17-s + 0.763·19-s − 7.23·23-s + 3.14·25-s + 0.381·29-s − 5.85·31-s + 5.29·35-s − 4·37-s + 7.23·41-s + 9.70·43-s − 8.94·47-s − 3.56·49-s − 11.8·53-s − 5.38·59-s − 14.1·61-s − 9.23·65-s + 8·67-s − 10.9·71-s + 14.3·73-s + 0.381·79-s + 1.61·83-s − 11.4·85-s − 6.76·89-s − 6·91-s + ⋯ |
L(s) = 1 | + 1.27·5-s + 0.700·7-s − 0.897·13-s − 0.970·17-s + 0.175·19-s − 1.50·23-s + 0.629·25-s + 0.0709·29-s − 1.05·31-s + 0.894·35-s − 0.657·37-s + 1.13·41-s + 1.48·43-s − 1.30·47-s − 0.508·49-s − 1.62·53-s − 0.700·59-s − 1.81·61-s − 1.14·65-s + 0.977·67-s − 1.29·71-s + 1.68·73-s + 0.0429·79-s + 0.177·83-s − 1.23·85-s − 0.716·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 1.85T + 7T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 + 7.23T + 23T^{2} \) |
| 29 | \( 1 - 0.381T + 29T^{2} \) |
| 31 | \( 1 + 5.85T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 7.23T + 41T^{2} \) |
| 43 | \( 1 - 9.70T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 + 5.38T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 0.381T + 79T^{2} \) |
| 83 | \( 1 - 1.61T + 83T^{2} \) |
| 89 | \( 1 + 6.76T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.57772829677167622295577763785, −6.54631061878400384083549435606, −6.09926747018263644527351919946, −5.32545176602096151736040971734, −4.75503490828726019898353598192, −3.99460593525872568221262852937, −2.83343224860032613475381927069, −2.05547890750408750945537378286, −1.57823508865137811035844110898, 0,
1.57823508865137811035844110898, 2.05547890750408750945537378286, 2.83343224860032613475381927069, 3.99460593525872568221262852937, 4.75503490828726019898353598192, 5.32545176602096151736040971734, 6.09926747018263644527351919946, 6.54631061878400384083549435606, 7.57772829677167622295577763785